Properties

Degree $1$
Conductor $83$
Sign $0.930 + 0.365i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.859 + 0.511i)2-s + (0.606 − 0.795i)3-s + (0.477 − 0.878i)4-s + (0.338 + 0.941i)5-s + (−0.114 + 0.993i)6-s + (−0.409 + 0.912i)7-s + (0.0383 + 0.999i)8-s + (−0.264 − 0.964i)9-s + (−0.771 − 0.636i)10-s + (0.953 − 0.301i)11-s + (−0.409 − 0.912i)12-s + (0.896 − 0.443i)13-s + (−0.114 − 0.993i)14-s + (0.953 + 0.301i)15-s + (−0.543 − 0.839i)16-s + (0.190 + 0.981i)17-s + ⋯
L(s,χ)  = 1  + (−0.859 + 0.511i)2-s + (0.606 − 0.795i)3-s + (0.477 − 0.878i)4-s + (0.338 + 0.941i)5-s + (−0.114 + 0.993i)6-s + (−0.409 + 0.912i)7-s + (0.0383 + 0.999i)8-s + (−0.264 − 0.964i)9-s + (−0.771 − 0.636i)10-s + (0.953 − 0.301i)11-s + (−0.409 − 0.912i)12-s + (0.896 − 0.443i)13-s + (−0.114 − 0.993i)14-s + (0.953 + 0.301i)15-s + (−0.543 − 0.839i)16-s + (0.190 + 0.981i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.930 + 0.365i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.930 + 0.365i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $0.930 + 0.365i$
Motivic weight: \(0\)
Character: $\chi_{83} (59, \cdot )$
Sato-Tate group: $\mu(41)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (0:\ ),\ 0.930 + 0.365i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.8284545687 + 0.1569137263i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.8284545687 + 0.1569137263i\)
\(L(\chi,1)\) \(\approx\) \(0.8863160301 + 0.1149247595i\)
\(L(1,\chi)\) \(\approx\) \(0.8863160301 + 0.1149247595i\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−30.68981172557789931521211400639, −29.59577547138028421150757757482, −28.49217581658478108109895208157, −27.57392459915070623191846100021, −26.79898478312650712562425400548, −25.52836254988520366357397813214, −25.09009617394740562460691133159, −23.217646508601697156293424286718, −21.68005984075781301028933265697, −20.84961007024976229739135859162, −20.03390218613513687500351139232, −19.27001707868982476584979251899, −17.49711527555476516428141897130, −16.60732200213054531055655021980, −15.87515109799105082968924695569, −14.0366939575802778733055197613, −12.96381845864159745724188151057, −11.42653202155048730241225665709, −10.15020024024779880026162461878, −9.26003666814773112881735007015, −8.4349799239464733028410663617, −6.82569174809001197594338838614, −4.54325829449381382048550706836, −3.403415146834937748357378363728, −1.49899238978799929824132095013, 1.73785948284241891706597648708, 3.114426284616928924983249849415, 6.197487664601867955645505483720, 6.49583809412489378257352402417, 8.17945708781522495487030192084, 9.00072336369753331486026623533, 10.372882574580528421612452403562, 11.76418031006156841782417894350, 13.342536118151844158799717247159, 14.687355221613104177358660331320, 15.24211343807081541941978264686, 16.97148145536815409674820875270, 18.09297538468444440134978633931, 18.91775022771256877019585935613, 19.49055456507099899031565524972, 21.04915614126940022387281227809, 22.56915471064186519231028450406, 23.75577725612191294990788721912, 25.083356001144208028852888406537, 25.468501385342954030768154004107, 26.387612363215750239995078883524, 27.65082255051765202058657572426, 28.83195262203088038829462752044, 29.86165249743207199514135562556, 30.6865347211182201686707846134

Graph of the $Z$-function along the critical line